"Hashem appeared to Avram and said, 'To your offspring I will give this land.' So he built an altar [in Canaan] to Hashem Who appeared to him. From there he relocated to the mountain east of Beth-el and pitched his tent, with Beth-el on the west and Ai on the east; and he built there an altar to Hashem and invoked Hashem by Name. Then Avram journeyed on, going and traveling toward the south. There was a famine in the land, and Avram descended to Egypt to sojourn there, for the famine was severe in the land." ~Bereishit 12;7-10
"So Avram went up from Egypt, he with his wife and all that was his- and Lot with him- to the south. Now Avram was very heavy with livestock, with silver, and with gold. He proceeded on his journeys from the south to Beth-el, to the place where his tent had been at first, between Beth-el and Ai, to the site of the altar which he had made there at first; and there Avram invoked Hashem by Name." ~Bereishit 13;1-2
Vectors:
In their introductory to Geometry, younger students are introduced to the concepts of points, lines, line segments, and rays.
*A point is a single point or location.
*A line is a straight connection between two points that continues straight and extends indefinitely in both directions
*A line segment is the straight connection between two points with endpoints at the two points
*A ray is the straight connection between two points with an endpoint at one of the points and extending indefinitely through the other point
A vector is a concept that is not classically introduced to younger students. Essentially, a vector is the same idea as a segment, but it also includes information about distance and tells which direction you're moving in. The two key aspects to a vector are distance (referred to as "magnitude") and direction. Velocity explains the speed at which something is moving (again, it's magnitude) and the direction in which it is moving (23 miles per hour NorthEast). With this understanding of vector and velocity, we can see that velocity is a type of vector measurement, and we can use this understanding to measure how far something travels and the direction in which it is traveling. One simple example would be calculating how far something is traveling either towards (+) or away from (-) a given point. To think about vectors in their simplest form, every straight movement in one direction can be shown as one vector. This means that if I'm traveling in one direction, and then I change my direction, I would need to use 2 different vectors- 1 to explain the first part of my trip, and 1 to explain my trip after I change my direction.
The vector label is based on directions relative to a specific reference point. It's important to realize that this information only tells you have far and in what direction you've moved from your starting point; by itself it doesn't tell you how close or far you are from your reference point. In order to know where you are related to your reference point, you need to add or subtract different vectors to calculate the distance. Using the calculations of multiple vectors, this type of information can be charted on graphs showing distance compared to a location over time on a coordinate plane, and adding the aspect of direction adds an additional quality of information to the graph. For today, we'll just look at simple vectors.
Parsha Connection:
In this week's parsha, we learn of Avram's travels over the course of 25 years. If we look specifically at his travel from the beginning of the 12th chapter through the beginning of the 13th chapter of Bereishit, we see him set-up his tent in Canaan between Beth-el and Ai, travel down to Egypt for the duration of a famine, and then travel back up to return to the exact same location between Beth-el and Ai. How can we use vectors or velocity to explain his trip?
Based on a rough estimate found here, let's assume that Avram's travel distance for his trip from Beth-el down to Egypt was approximately 225 miles. If we designate Beth-el as his starting point (the location where he wants to be) and we use the standard of North and East representing positive directions and South and West representing negative, then his trip down to Egypt is mileage expressed as a negative velocity. So, for example, he starts at 0 miles (when he's standing in Beth-el), and after traveling 100 miles towards Egypt, he had traveled 100 miles SouthWest, or -100 miles, because he was moving away from Beth-el in a negative direction. When he reached his final destination in Egypt, he had traveled 225 miles SouthWest, or -225 miles. On his return trip, he is now heading NorthEast, which is a positive direction. As he traveled back toward Beth-el, let's say he traveled 60 miles per day (velocity = 60 mph NE)- assuming he was traveling by camel.
*After 1 day, he was 60 miles closer to Beth-el, so he had traveled 60 miles NorthEast, or +60 miles
*After 2 days, he was another 60 miles closer, so he had traveled 120 miles NorthEast, or +120 miles
*After 3 days, he was another 60 miles closer, so he had traveled 180 miles NorthEast, or +180 miles
*On his 4th day, he covered the final 45 mi to reach his destination, and he stopped once he reached his 225 miles NorthEast, or +225 miles, to get to Beth-el.
Note that you can see here that by adding the individual vectors from each day, we get an accumulated larger vector, only because Avram was continuing his travel in the exact same direction.
Extension Thoughts:
If we add the two vectors from each trip together, we get zero - he's back to his initial starting point. (-225 miles) + (+225 miles) = 0 miles. If we add the vectors after the first day of Avram's return trip, his total travels still end up as a negative vector. He travelled 225 miles SouthWest, and then 60 miles NorthEast. (-225 miles) + (+60 miles) = (-165 miles), or 165 miles SouthWest of the vector's starting point.
Everyday Activity:
The concept of vectors can be quite easily integrated into younger students learning about measurement and distance. Simple classroom activities can include students measuring distance from a reference point and then having them walk certain distances in directions related to your reference point and expressing their walks using vector language.
A simple everyday example of vectors: Have you ever paid attention to the highway distance markers as you drive along? Using those markers, you can say that you've driven 23 miles North toward your destination.
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